1. Field of the Invention
The present invention relates to finite difference algorithms and associated models for viscoelastic ink ejection simulation.
2. Description of the Related Art
Results of ink-jet simulations are useful in the design of piezoelectric ink-jet print heads. A practical ink-jet simulation may be carried out using an analytical tool such as an equivalent circuit that receives as an input the dynamic voltage to be applied to a piezoelectric actuator and simulates the ink behavior under the influence of the ink cartridge, supply channel, vibration plate, and actuator. That is, from the input voltage and an ink flow rate, the equivalent circuit calculates an inflow pressure that drives computational fluid dynamics (CFD) code. The CFD code then solves the governing partial differential equations, i.e., the incompressible Navier-Stokes equations for two-phase flows, for fluid velocity, pressure, and interface position, and feeds back the ink flow rate to the equivalent circuit. The sequence is repeated as long as needed.
The dynamics of fluid flow through the ink-jet nozzle, coupled to surface tension effects along the ink-air interface and boundary conditions along the wall, act to determine the shape of the interface as it moves. Designing the CFD code largely involves taking into account the dynamically changing ink-air interface, which can be quite challenging.
One method that has been used to model the ink-air interface is the volume of fluid method (VOF). The VOF method performs mass conservation fairly well but has problem accurately modeling surface tension aspects of fluid flow, especially when the ink droplet is very small, as in less than 5 picoliters. The ability to eject small ink droplets is essential for photo quality ink-jet printers. The VOF method has given way to improved modeling methods, which include level set methods that are better at accurately capturing the ink-air interface in CFD simulations than the VOF method. There is an explicit relationship between the level set, the interface curvature, and the surface tension. This relationship allows the level set method to excel whenever surface tension is important.
Level set methods may make use of the finite element method. One problem with using the finite element method when applied to the level set method is the inability of the finite element method to accurately and effectively conserve the mass of the fluid being simulated. Finite difference analysis is better at conserving the mass of the fluid being simulated.
The dynamics of fluid flow are further complicated when the fluid is viscoelastic in nature. A viscoelastic fluid may be characterized by several parameters including, density, viscosity, elastic modulus, and others. A variety of numerical models are used to describe viscoelastic fluids including the Maxwell model, the Kelvin-Voigt model, Generalized Maxwell Model, the Standard Linear Solid Model, and the Oldroyd-B model.
The viscosity and the elastic modulus may be combined to give a relaxation time. The relaxation time of a viscoelastic fluid characterizes the amount of time it takes the fluid to dissipate elastic energy stored in the fluid. The relaxation time is normalized relative to the time period of the observation of the numerical experiment. The normalized relaxation time is referred to as the Deborah number. Prior art methods that attempted simulate the jettability of a viscoelastic fluid would only work when the Deborah number was less than 3.
The present invention is directed towards a method of simulating a viscoelastic fluid when the Deborah number is greater than 3.